A2. Use Bordered Hessians to determine the sign properties (definiteness) of the following con- strained quadratic forms: (i) Q(x₁, x2, x3) = −x² + x² + x² + 4x1x2 – 2x1x3 subject to x₁+x2+x3 = 0 and −x1+x2+x3 = 0, (ii) Q(x, y, z) = −x² + y² − 6xz + 4yz subject to −x + y + z = 0.

4. Find the roots of the following over C (a) x2 + 18x+97 (b) x3 + 3x2 – 6x-8 yes that is a cubic (c) 2x2 - 17x - 31 (d) 12x3 - 32x2 + 11x + 15 yes that is also a cubic

7. List all the possible Q roots, do NOT factor. (a) f(x) = x - 4r² +1 (b) g(x) = 4r3 – 4r2 – 17x – 7 (c) f(r) = 3.r3 – 4x2 + 5x – 9 (d) f(x) = 3x4 + 3x3 + 3r2 – 20

Solve 2.

Examine whether the following quadratic forms are positive definite, negative definite or indefinite.

Consider the following quadratic form on R²: 43 2 57 2 + -x7² 25 -X₂ 2 25 Q(x) = [₁] = 48 a) Determine whether Q is positive or negative (semi)definite. b) Find a change of variables x = Py that converts Q to a form with no cross-product terms. c) write Q in terms of the new variables y 1 and y2. In your formula, write 'y_1' and 'y_2' for y₁ and y2. Use the square root symbol 'V' where needed to give an exact value for your answer. [00₁] 0 0 1 00 42 +. a) The quadratic form is c) Q(x) = 0 25 -X1X2 Positive Definite Negative Definite Positive Semidefinite Negative Semidefinite Indefinite